Basic Concepts
The M/M/s/N queueing model is the same as the M/M/s model, except that the customer population is finite, with N members. Alternatively, we can view the model as similar to the M/M/1/N model, except that there are s servers.
Properties
Using the notation found in Queueing Theory, we now present formulas for the key properties of this queueing model once a steady state is reached. As usual, we set τ = λ/μ.
For n ≤ s
For s ≤ n ≤ N
Of course, for n > N, pn = 0. Also
where
By Little’s Law
Example
Example 1: Calculate L, Lq, W, Wq, and pn for the M/M/s/N model with λ, μ, s, and N specified in column B of Figure 1.
The values for L, Lq, W, Wq, ρ, τ, and λ-bar are calculated on the right side of Figure 1. Column K contains a list of the formulas used in column G. The formula in cell G2 makes reference to cells in Figure 2, shown below.
Figure 1 – M/M/s/N queueing model (part 1)
Figure 2 shows how to calculate the pn = the probability that n customers are in the system and Pn = the probability that at most n customers are in the system.
p0 in cell N3 is calculated via the formula =1/SUM(R3:R23). Here cell R3 contains the formula
=IF(Q3>B$5,0,(G$8/B$5)^Q3*COMBIN(B$5,Q3)*IF(Q3<B$4,1,PERMUT(Q3,Q3-B$4)/B$4^(Q3-B$4)))
The other formulas in column R are inserted by highlighting range R3:R23 and pressing Ctrl-D. Cell N4 contains the formula =N$3*R4. The other formulas in column N are inserted in a similar fashion by highlighting N4:N23 and pressing Ctrl-D.
Note that column R contains values for n = 0 to 20 (only values n = 0 to 12 are displayed). If N > 20 then we need to expand range R3:R23 to at least n = N. The situation is similar for column N.
Figure 2 – M/M/s/N queueing model (part 2)
Worksheet Functions
Excel Functions: The Real Statistics Resource Pack supports the following array function.
MMsN(λ, μ, s, nn, lab): returns a column array with the L, Lq, W, Wq, λ-bar in the steady state for the M/M/s/N queueing model with exponential arrival and service rates with mean arrival rate λ and mean service rate μ, and with s servers and a population of size nn.
If lab = TRUE (default FALSE) then an extra column of labels is appended to the output. In addition, the following non-array functions are also supported.
MMsNPn(λ, μ, s, nn, n, cum) = pn for the M/M/s/N queueing defined by s, nn, λ and μ if cum = FALSE (default) and Pn otherwise.
We can obtain the values in range F2:G6 of Figure 1 via the array formula =MMsN(B2,B3,B4,B5,TRUE). We can obtain the values in Figure 2 by inserting =MMsNPn(B$2,B$3,B$4,B$5,M3) in cell N3 and =MMsNPn(B$2,B$3,B$4,B$5,M3,TRUE) in cell O3, highlighting N3:O15, and pressing Ctrl-D.
Links
Examples Workbook
Click here to download the Excel workbook with the examples described on this webpage.
References
Ross, S. M. (2014) Introduction to probability models, 11th Ed. Academic Press
https://ebin.pub/introduction-to-probability-models-11nbsped-0124079482-9780124079489.html
Sztrik, J. (2021) Basic queueing theory
https://irh.inf.unideb.hu/~jsztrik/education/16/SOR_Main_Angol.pdf
Shores, T. S. (2017) Queueing theory basics and models
No longer available online